Question: $ F = \left[\begin{array}{rr}2 & 3 \\ 4 & -2 \\ -2 & 2\end{array}\right]$ $ E = \left[\begin{array}{rr}-1 & -1 \\ 4 & -2\end{array}\right]$ What is $ F E$ ?
Solution: Because $ F$ has dimensions $(3\times2)$ and $ E$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ F E = \left[\begin{array}{rr}{2} & {3} \\ {4} & {-2} \\ \color{gray}{-2} & \color{gray}{2}\end{array}\right] \left[\begin{array}{rr}{-1} & \color{#DF0030}{-1} \\ {4} & \color{#DF0030}{-2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rr}{2}\cdot{-1}+{3}\cdot{4} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{2}\cdot{-1}+{3}\cdot{4} & ? \\ {4}\cdot{-1}+{-2}\cdot{4} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ F$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{2}\cdot{-1}+{3}\cdot{4} & {2}\cdot\color{#DF0030}{-1}+{3}\cdot\color{#DF0030}{-2} \\ {4}\cdot{-1}+{-2}\cdot{4} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{2}\cdot{-1}+{3}\cdot{4} & {2}\cdot\color{#DF0030}{-1}+{3}\cdot\color{#DF0030}{-2} \\ {4}\cdot{-1}+{-2}\cdot{4} & {4}\cdot\color{#DF0030}{-1}+{-2}\cdot\color{#DF0030}{-2} \\ \color{gray}{-2}\cdot{-1}+\color{gray}{2}\cdot{4} & \color{gray}{-2}\cdot\color{#DF0030}{-1}+\color{gray}{2}\cdot\color{#DF0030}{-2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}10 & -8 \\ -12 & 0 \\ 10 & -2\end{array}\right] $